Section 6.6 Vectors 4/26/2017

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HPC/RPC 2017

https://www.youtube.com/watch?v=A05n32Bl0aY

(a,b)•

Two – Dimensional Vectors While (a,b) denotes a point on a plane it also represents a directed line segment with its tail (initial point) at the origin (0,0) and the head (terminal point) at (a,b).The vector has both magnitude and directionand is denoted as , in order to distinguish from point (a,b).

v = ,

(0,0)

y

x

Head

Tail

Section 6.6 Vectors 4/26/2017

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DEFINITION: Two-Dimensional VectorA 2-D vector, v, is an ordered pair of real numbers in component form , .The numbers a, b are the components of the vector v.The standard representation (also known as “component form”) of vector , is the arrow from the origin (0,0) to point (a,b). The magnitude of v is the length of the line segment.The direction of v is the direction (given as an angle measure) that the arrow is pointing.Zero vector, 0 = 0, 0 has no length or direction

P1

P2

Initial Point

Terminal Point

22 , yx

11, yx

Section 6.6 Vectors 4/26/2017

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Component Form of a VectorVectors can be compared (magnitude and direction) as long as they are in component form.Head Minus Tail (HMT) Rule to find Component Form:If a vector has initial point (x1, y1) and terminal point (x2, y2),it represents the vector ,Example 1: Find component form of vectors u and v.u has initial point (1, 1) and terminal point (2, 4)

v has initial point (5, 3) and terminal point (6, 6)<2 – 1, 4 – 1> = <1, 3>

<6 – 5, 6 – 3> = <1, 3>

• If v is represented by the arrow from (x1,y1) to (x2, y2) then the magnitude of the vector is represented by:

• If the vector is in Component Form , , the magnitudecan be found using

Section 6.6 Vectors 4/26/2017

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MagnitudeExample 1: Find the magnitude of vectors u and v.u has initial point (1, 1) and terminal point (2, 4)

v has initial point (5, 3) and terminal point (6, 6)

2 1 4 1

6 5 6 3

1 31 9

1 31 9

Vector OperationsVector Additionif u = , and

v = , ,the sum of vectors u andv is the vector:u + v = ,(This is also know as the“resultant vector”)

Scalar Multiplicationif u = , andk is a real number(scalar) the product ofthe scalar k and thevector u is:ku = k ,,

Section 6.6 Vectors 4/26/2017

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The negative of a vector is just a vector going the opposite way.

v vA number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

v v vv3

Scalar MultiplicationA vector can be multiplied by a real number k. If k is a positive number then the magnitude of the

vector is changed If k is a negative number then the magnitude is changed

and the direction is reversed.

u 2u u21 u2

3

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To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

v w

Initial point of v vw

Move w over keeping the magnitude and direction the same.

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words).

Terminal point of w

w

u v w vu uvw3

w ww

Using the vectors shown, find the following:

vu u v

vwu 32u u w w w

v

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Example 3: Performing Vector OperationsIf u = 3, 2 and v = 7, 3 find the following:

a. u + v =

b. 3v =

c. 2u + (-1)v

<-3 + 7, 2 + 3> = <4, 5>

3 <7, 3> = <21, 9>

=2 <-3, 2> + (-1) <7, 3>=<-6, 4> + <-7, -3>= <-13, 1>